Radiation stress

Definition of Radiation stress: Radiation stress is the excess flux of momentum carried by ocean waves. This is the common definition for Radiation stress, other definitions can be discussed in the article

Notes

The fact that spatial gradients in wave energy can induce a net force on the water body was first recognised by Longuet-Higgins and Stewart (1962, 1964). The radiation stresses are the elements of a stress tensor representing the phase-averaged and depth-integrated excess momentum flux through vertical planes due to horizontal wave orbital motion and the wave-induced pressure.

The radiation stresses are given by (see Shallow-water wave theory#Radiation Stress (Momentum Flux))

[math]S_{XX} =\Bigl\langle \int _{-h}^{\eta } (p+\rho u^{2} )dz \Bigr\rangle - \int _{-h}^{0} p_0 dz , \qquad S_{YY} =\Bigl\langle \int _{-h}^{\eta }(p+\rho v^{2} )dz \Bigr\rangle -\int _{-h}^{0}p_0 dz , \qquad S_{XY} =\Bigl\langle \int _{-h}^{\eta } \rho u v dz \Bigr\rangle , \qquad (1)[/math]

where [math]z=[/math] vertical coordinate, [math]h=[/math] water depth, [math]\eta =[/math] wave surface elevation, [math]\rho = [/math] water density, [math]p =[/math] pressure, [math]p_0 =[/math] hydrostatic pressure, [math]u=[/math] horizontal orbital velocity in [math]x[/math]-direction, [math]v=[/math] horizontal orbital velocity in [math]y[/math]-direction, and where [math] \bigl\langle … \bigr\rangle [/math] represents the average over the wave period.

For obliquely incident waves, cross-shore and longshore wave orbital motions contribute to transferring mean momentum in both cross-shore and longshore directions. Spatial variation of the wave field, for example due to wave breaking, therefore generates gradients in the radiation stress that exert a mean force on the water mass. The cross-shore component of this force generates a water level set-up at the coast, whereas the longshore component generates a longshore current. Radiation stress gradients associated with wave breaking generally provide the dominant forcing of wave-induced mean water levels and currents in the surf zone.

Approximate analytical expressions of the radiation stresses can be obtained from linear wave theory. Assuming a uniform wave field propagating with an angle [math]\theta[/math] to the cross-shore [math]x[/math]-axis the expressions are

[math]S_{XX} \approx \Big(n(1+\cos^2 \theta) - \large\frac{1}{2}\normalsize \Big) E , \qquad S_{YY} \approx \Big(n(1+\sin^2 \theta) - \large\frac{1}{2}\normalsize \Big) E , \qquad S_{XY} \approx nE \sin \theta \cos \theta , \qquad (2) [/math]

where [math]E = \large\frac{1}{8}\normalsize \rho g H^2 =[/math] wave energy, [math]H=[/math] wave height, [math]k=[/math] wave number, [math]g=[/math] gravitational acceleration, and [math]n = \large\frac{1}{2}\normalsize + \Large\frac{kh}{\sinh (2kh)}\normalsize =[/math] ratio of group celerity and wave celerity.

These expressions underestimate the radiation stresses when waves become skewed and asymmetric while propagating into the shoaling zone - see Shallow-water wave theory#Finite amplitude waves. For example, calculations with 5th order nonlinear Stokes theory give a 17% higher value of [math]S_{XX}[/math] in the case of very steep waves. On the other hand, the linear wave expressions (2) overestimate the radiation stresses in the surf zone.

For irrotational periodic (regular) gravity waves an exact expression of [math]S_{XX}[/math] is given by

[math]S_{XX} = 4 E_k - 3 E_p + \rho h \bigl\langle u_b^2 \bigr\rangle , \quad E_k = \large\frac{1}{2}\normalsize \rho \Bigl\langle \int_{-h}^{\eta} (u^2+w^2)dz \Bigr\rangle , \quad E_p = \large\frac{1}{2}\normalsize \rho g \bigl\langle \eta^2 \bigr\rangle , [/math]

where [math]u=[/math] cross-shore wave orbital velocity, [math]u_b=[/math] cross-shore wave orbital velocity at the bottom, [math]w=[/math] vertical wave orbital velocity.

Related articles

Shallow-water wave theory

Wave set-up

Undertow

Breaker index

Shoreface profile